Michael Shulman
discrete object

An object AA of a 2-category KK is discrete if the category K(X,A)K(X,A) is equivalent to a discrete set for all objects XX of KK. Discrete objects are also called 0-truncated objects since they are characterized by K(X,A)K(X,A) being a 0-category (a set).

More explicitly, an object AA is discrete if and only if every pair of parallel 2-cells α,β:f⇉g:X⇉A\alpha,\beta:f \;\rightrightarrows\; g:X\;\rightrightarrows\;A are equal and invertible. If KK has finite limits, this can be expressed equivalently by saying that A→ApprA\to A^{ppr} is an equivalence, where pprppr is the “walking parallel pair of arrows.”

We write disc(K)disc(K) for the full sub-2-category of KK on the discrete objects; it is equivalent to a 1-category, and is closed under limits in KK.

A morphism A→BA\to B is called discrete if it is discrete as an object of the slice 2-categoryK/BK/B.